Right-triangle revision | Year 11

01 / LEARN

Understand the idea.

Label before calculating

Choose the angle before naming its sides. The hypotenuse is always opposite the right angle; the opposite leg faces your chosen angle, while the adjacent leg touches it. Changing the chosen angle swaps the two leg labels.

\begin{aligned} sin θ & = \frac{o}{h} \\ cos θ & = \frac{a}{h} \\ tan θ & = \frac{o}{a} \end{aligned}

Solve and check

Select the ratio containing the side you know and the quantity you want. Rearrange the equation before entering numbers, and use an inverse trigonometric function when the unknown is an angle.

For a degree question, check degree mode. Round only the final result, then check the geometry: an acute angle lies between $0^{\circ}$ and $90^{\circ}$ , and neither leg can exceed the hypotenuse.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

A 5 m ladder makes an angle of $60^{\circ}$ with level ground. Find its vertical height.

Worked solution

1
Identify the opposite side and hypotenuse.
$sin 60^{\circ} = \frac{h}{5}$
The vertical height faces the ground angle, and the ladder is the hypotenuse.
2
Multiply by the ladder length.
$h = 5 sin 60^{\circ}$
Multiplying both sides by 5 isolates the height.
3
Evaluate exactly, then round.
$h = \frac{5 \sqrt{3}}{2} m \approx 4.33 m$
The exact value is $sin 60^{\circ} = \frac{\sqrt{3}}{2}$ ; the height is less than 5 m.

Answer

$5 \sqrt{3} / 2 m \approx 4.33 m$

EXAMPLE 02

Worked example 2

The legs opposite and adjacent to theta are 6 cm and 8 cm. Find theta to one decimal place.

Worked solution

1
Use the ratio of the two legs.
$tan θ = \frac{6}{8} = \frac{3}{4}$
Tangent compares opposite with adjacent and needs no hypotenuse.
2
Apply inverse tangent in degree mode.
$θ = {tan}^{- 1} (\frac{3}{4}) \approx {36.8699}^{\circ}$
Inverse tangent returns the acute angle with this leg ratio.
3
Round to one decimal place.
$θ \approx {36.9}^{\circ}$
The result lies between $0^{\circ}$ and $90^{\circ}$ , as an acute triangle angle must.

Answer

${36.9}^{\circ}$

03 / YOUR TURN

Try it yourself.

Check your understanding

Opposite is 5 and hypotenuse is 13. Find $sin θ$ .

Show answer & reasoning

$5 / 13$

Sine compares the opposite side with the hypotenuse, so $sin θ = \frac{5}{13}$ . The ratio is less than 1, as expected.

Keep practising

6 worksheet questions · 18 marks

Open worksheet Worked solutions

Read the questions online

A right triangle has legs 9 cm and 12 cm. For the angle opposite 9 cm, find all three ratios.
3 marks
Find the opposite side when the hypotenuse is 14 m and the angle is $30^{\circ}$ .
3 marks
The adjacent leg is 8 cm and the angle is $60^{\circ}$ . Find the hypotenuse.
3 marks
A ramp rises 1.2 m over a horizontal run of 4 m. Find its angle to one decimal place.
3 marks
A wire makes $35^{\circ}$ with horizontal ground and reaches 10 m horizontally. Find its length to two decimal places.
3 marks
Explain why a right triangle cannot have hypotenuse 7 cm and leg 9 cm.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A right triangle has legs 9 cm and 12 cm. For the angle opposite 9 cm, find all three ratios.
3 marks
$h = \sqrt{9^{2} + 12^{2}} = 15$ .
Substitute into the definitions and simplify.
$sin θ = \frac{3}{5}, cos θ = \frac{4}{5}, tan θ = \frac{3}{4}$
Find the opposite side when the hypotenuse is 14 m and the angle is $30^{\circ}$ .
3 marks
$o = 14 sin 30^{\circ} = 7$ .
7 m
The adjacent leg is 8 cm and the angle is $60^{\circ}$ . Find the hypotenuse.
3 marks
$cos 60^{\circ} = \frac{8}{h}$ , so $h = 16$ .
16 cm
A ramp rises 1.2 m over a horizontal run of 4 m. Find its angle to one decimal place.
3 marks
$tan θ = \frac{1.2}{4} = 0.3$ .
$θ = {tan}^{- 1} (0.3) \approx {16.699}^{\circ}$ .
${16.7}^{\circ}$
A wire makes $35^{\circ}$ with horizontal ground and reaches 10 m horizontally. Find its length to two decimal places.
3 marks
$L = \frac{10}{cos 35^{\circ}} \approx 12.2077$ .
12.21 m
Explain why a right triangle cannot have hypotenuse 7 cm and leg 9 cm.
3 marks
A sine or cosine ratio of $\frac{9}{7}$ would exceed 1, confirming inconsistent side data.
The hypotenuse must be longest.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: triangle rules, circular measure and trigonometric functions; school sequencing may vary.

Official curriculum source ↗

In this lesson

Understand the idea.

Label before calculating

Solve and check

See the method in action.

Worked example 1

Identify the opposite side and hypotenuse.

Multiply by the ladder length.

Evaluate exactly, then round.

Worked example 2

Use the ratio of the two legs.

Apply inverse tangent in degree mode.

Round to one decimal place.

Try it yourself.

Check your understanding

Keep practising